Asymptotic approach for the rigid condition of appearance of the oscillations in the solution of the Painleve-2 equation

The asymptotic solution for the Painleve-2 equation with small parameter is considered. The solution has algebraic behavior before point t∗ and fast oscillating behavior after the point t∗. In the transition layer the behavior of the asymptotic solution is more complicated. The leading term of the asymptotics satisfies the Painleve-1 equation and some elliptic equation with constant coefficients, where the solution of the Painleve-1 equation has poles. The uniform smooth asymptotics are constructed in the interval, containing the critical point t∗. In this work a special asymptotic solution on ε at ε → 0 for the equation Painleve-2 is constructed: εu + 2u + tu = 1. (1) The constructed solution describes a rigid condition of origin of fast oscillations in some moment t∗. At the left, at t < t∗, the asymptotic solution is algebraic, and on the right, at t > t∗, this asymptotics is fast oscillating. An asymptotic solution in a transitional layer (small neighborhood of a critical point t∗) is investigated explicitly. The phase and phase shift of the oscillating asymptotics are calculated. The qualitative behaviour of solutions of second-order ordinary differential equations according to an additional parameter is explained, for example, in the book [1]. In [1] the various types of bifurcations for equilibrium positions of conservative second-order ordinary differential equations are described also. Consider the autonomous equation obtained from (1). If we ”freeze” a value of variable coefficient t = T then we obtain the equation: εV ′′ + 2V 3 + TV = 1. (2)